Subgroups — #30827339

Order 17152.

644 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569227	C₂	1	2	yes	maximal, complemented
2	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
3	#22559124	C₂	1	2	yes	maximal, complemented
4	#22559089	C₂	1	4	yes	maximal, complemented
5	#22559029	C₂	1	2	yes	maximal, complemented
6	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558644	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₄	1	1	yes	characteristic
10	C₄×#11761628	C₂²	1	2	yes	—
11	C₈×Dic₁₃₄	C₂²	1	4	yes	—
12	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic
13	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
14	C₄×#11761617	C₄	1	2	yes	—
15	C₄×#11761617	C₂²	1	2	yes	—
16	C₄×D₅₃₆	C₄	1	1	yes	characteristic
17	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
18	#16863475	C₂²	1	4	yes	—
19	#16863472	—	2	4	no	—
20	#16863453	C₂²	1	2	yes	—
21	#16863451	—	2	2	no	—
22	#16863434	C₄	1	1	yes	characteristic
23	#16863393	C₄	1	2	yes	—
24	#16863348	—	2	2	no	—
25	#16863149	C₂²	1	2	yes	—
26	#16863147	C₂²	1	2	yes	—
27	#16863128	C₂²	1	4	yes	—
28	#16863125	C₂²	1	4	yes	—
29	#431×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#322×C₆₇	C₂²	1	2	yes	—
33	#322×C₆₇	C₂²	1	2	yes	—
34	C₂×#11761628	C₂×C₄	1	2	yes	—
35	#14524065	—	2	4	no	—
36	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
37	C₂×#11761617	C₂×C₄	1	2	yes	—
38	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
39	C₂×#11761611	—	2	4	no	—
40	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₃₄	C₂³	1	2	yes	—
42	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
43	C₄×D₂₆₈	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₆₇	—	2	2	no	—
45	C₄×C₂²⋊D₆₇	—	2	2	no	—
46	#14524002	C₂×C₄	1	2	yes	—
47	#14524000	—	2	2	no	—
48	C₄²×D₆₇	—	2	2	no	—
49	#14523977	C₂×C₄	1	1	yes	characteristic
50	#14523977	C₂×C₄	1	1	yes	characteristic
51	#14523975	C₂×C₄	1	2	yes	—
52	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
53	Dic₆₇⋊C₈	—	2	4	no	—
54	#14523968	C₂×C₄	1	2	yes	—
55	#14523968	C₂×C₄	1	2	yes	—
56	#14523965	C₂×C₄	1	2	yes	—
57	#14523965	C₂×C₄	1	2	yes	—
58	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
59	(C₂×D₆₇)⋊C₈	—	2	4	no	—
60	#14523954	—	2	2	no	—
61	C₈×Dic₆₇	—	2	4	no	—
62	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
63	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₆₇	—	2	1	no	—
67	C₂²⋊C₈×C₆₇	—	2	1	no	—
68	#97×C₆₇	D₄	1	2	yes	—
69	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₆₇	C₂³	1	2	yes	—
72	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	2	yes	—
75	#11761628	—	2	4	no	—
76	#11761628	—	4	2	no	—
77	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
78	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
79	C₂×C₂²⋊D₆₇	—	2	2	no	—
80	C₂×C₄×D₆₇	—	2	2	no	—
81	C₂×C₄×D₆₇	—	2	2	no	—
82	C₂×C₄×D₆₇	—	2	2	no	—
83	Dic₂₆₈	M₁₆	1	4	yes	complemented
84	#11761617	—	2	4	no	—
85	D₅₃₆	—	2	2	no	—
86	#11761611	—	4	8	no	—
87	C₂×C₆₇⋊C₈	—	2	4	no	—
88	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
89	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
90	Dic₆₇⋊C₄	—	2	2	no	—
91	Dic₆₇⋊C₄	—	2	2	no	—
92	C₂²⋊Dic₆₇	—	2	2	no	—
93	(C₂×D₆₇)⋊C₄	—	2	2	no	—
94	(C₂×D₆₇)⋊C₄	—	2	2	no	—
95	C₄×Dic₆₇	—	2	2	no	—
96	C₄×Dic₆₇	—	2	2	no	—
97	C₄×Dic₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
101	M₁₆×C₆₇	—	4	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	4	1	no	—
105	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
110	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₇	#41	1	1	yes	characteristic
113	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
115	C₂²×D₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	4	1	no	—
120	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
126	C₂²⋊D₆₇	—	4	4	no	—
127	C₂×Dic₆₇	—	2	2	no	—
128	C₂×Dic₆₇	—	2	2	no	—
129	C₂×Dic₆₇	—	2	2	no	—
130	D₂₆₈	—	2	2	no	—
131	D₂₆₈	—	2	2	no	—
132	C₄×D₆₇	—	4	4	no	—
133	C₄×D₆₇	—	4	4	no	—
134	C₄×D₆₇	—	8	2	no	—
135	Dic₁₃₄	—	2	2	no	—
136	Dic₁₃₄	C₂×M₁₆	1	4	yes	—
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	—	2	1	no	—
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	2	4	no	—
141	C₈×C₆₇	—	4	2	no	—
142	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
143	C₆₇⋊C₈	—	4	4	no	—
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×D₆₇	—	4	2	no	—
149	C₂×D₆₇	—	4	2	no	—
150	C₂×D₆₇	—	4	2	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
156	C₄×C₆₇	#568	1	1	yes	characteristic
157	C₄×C₆₇	#575	1	2	yes	—
158	Dic₆₇	—	2	4	no	—
159	Dic₆₇	—	4	2	no	—
160	Dic₆₇	—	4	2	no	—
161	#18043	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	—	4	1	no	—
163	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
164	C₂×C₆₇	#2943	1	1	yes	characteristic
165	C₂×C₆₇	#2957	1	1	yes	characteristic
166	D₆₇	—	8	2	no	—
167	#2971	—	67	2	no	—
168	C₄×#135	—	67	1	no	—
169	#1594	—	67	2	no	—
170	#1588	—	67	4	no	—
171	#1579	—	67	2	no	—
172	#1568	—	67	1	no	—
173	#1488	—	67	2	no	—
174	#1451	D₆₇	1	1	yes	characteristic, complemented
175	C₆₇	#18043	1	1	yes	characteristic, complemented, Sylow
176	C₂×#135	—	67	1	no	—
177	C₄×#41	—	67	2	no	—
178	C₈×Q₈	—	67	4	no	—
179	C₄×Q₁₆	—	67	1	no	—
180	C₄×Q₁₆	—	67	1	no	—
181	C₄×SD₁₆	—	67	2	no	—
182	C₄×SD₁₆	—	67	2	no	—
183	C₄×D₈	—	67	1	no	—
184	C₄×D₈	—	67	1	no	—
185	#436	—	67	4	no	—
186	#436	—	134	4	no	—
187	#432	—	67	2	no	—
188	#432	—	134	2	no	—
189	#431	C₂×D₆₇	1	1	yes	characteristic
190	#427	—	67	1	no	—
191	#416	—	67	2	no	—
192	C₄×M₁₆	—	134	2	no	—
193	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
195	#335	—	67	2	no	—
196	#334	—	67	2	no	—
197	Q₈⋊C₈	—	67	4	no	—
198	D₄⋊C₈	—	67	4	no	—
199	#322	C₂×D₆₇	1	2	yes	—
200	#322	C₂×D₆₇	1	2	yes	—
201	C₂×#41	—	67	2	no	—
202	#135	—	134	4	no	—
203	C₂×Q₁₆	—	67	1	no	—
204	C₂×SD₁₆	—	67	2	no	—
205	C₂×D₈	—	67	1	no	—
206	C₂×M₁₆	—	134	4	no	—
207	C₂×M₁₆	D₂₆₈	1	2	yes	—
208	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
209	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
210	C₄×Q₈	—	67	2	no	—
211	C₄×Q₈	—	67	2	no	—
212	C₄×D₄	—	67	2	no	—
213	C₄×D₄	—	67	2	no	—
214	C₄×D₄	—	134	2	no	—
215	C₄×D₄	—	134	2	no	—
216	#117	—	67	2	no	—
217	#117	—	134	2	no	—
218	C₂×C₄²	—	134	2	no	—
219	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
220	#107	—	67	1	no	—
221	#107	—	67	1	no	—
222	#106	—	67	2	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	67	4	no	—
225	C₄⋊C₈	—	134	4	no	—
226	#103	—	67	2	no	—
227	#103	—	67	2	no	—
228	#102	—	67	2	no	—
229	#102	—	67	2	no	—
230	C₂²⋊C₈	—	2	1	no	—
231	C₂²⋊C₈	—	134	2	no	—
232	C₂²⋊C₈	—	134	4	no	—
233	#97	—	134	2	no	—
234	#97	D₂₆₈	1	2	yes	—
235	C₄×C₈	—	134	4	no	—
236	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
237	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₆₇	1	2	yes	—
239	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₆₇	1	2	yes	—
242	#41	—	268	2	no	—
243	#41	—	134	4	no	—
244	C₂×Q₈	—	67	2	no	—
245	C₂×D₄	—	67	2	no	—
246	C₂×D₄	—	134	2	no	—
247	C₂²×C₄	—	134	2	no	—
248	C₂²×C₄	—	134	2	no	—
249	C₂²×C₄	—	134	2	no	—
250	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
252	C₂²×C₄	#11761628	1	1	yes	characteristic
253	Q₁₆	—	67	4	no	—
254	SD₁₆	—	134	4	no	—
255	D₈	—	134	2	no	—
256	M₁₆	—	4	2	no	—
257	M₁₆	—	268	8	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	134	4	no	—
262	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
263	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
267	C₄⋊C₄	—	67	2	no	—
268	C₄⋊C₄	—	67	2	no	—
269	C₄⋊C₄	—	134	2	no	—
270	C₄⋊C₄	—	134	2	no	—
271	C₂²⋊C₄	—	134	2	no	—
272	C₂²⋊C₄	—	134	2	no	—
273	C₂²⋊C₄	—	134	2	no	—
274	C₄²	—	134	2	no	—
275	C₄²	—	134	2	no	—
276	C₄²	—	134	2	no	—
277	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
278	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
279	C₄²	#11761628	1	1	yes	characteristic
280	C₄²	C₂³×D₆₇	1	1	yes	characteristic
281	C₂³	—	134	2	no	—
282	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
283	Q₈	—	67	4	no	—
284	Q₈	—	134	2	no	—
285	D₄	—	268	4	no	—
286	D₄	—	134	2	no	—
287	D₄	—	134	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	268	4	no	—
293	C₂×C₄	—	268	4	no	—
294	C₂×C₄	—	536	2	no	—
295	C₂×C₄	—	134	2	no	—
296	C₂×C₄	—	134	2	no	—
297	C₂×C₄	—	134	2	no	—
298	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
299	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
300	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
303	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	4	no	—
308	C₈	—	4	2	no	—
309	C₈	—	268	4	no	—
310	C₈	M₁₆×D₆₇	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	268	2	no	—
315	C₂²	—	268	2	no	—
316	C₂²	—	268	2	no	—
317	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	268	2	no	—
323	C₄	—	268	2	no	—
324	C₄	—	134	4	no	—
325	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
326	C₄	#16864162	1	1	yes	characteristic
327	C₄	#16864196	1	2	yes	—
328	C₂	—	4	1	no	—
329	C₂	—	536	2	no	—
330	C₂	M₁₆×D₂₆₈	1	1	yes	characteristic
331	C₂	#22569011	1	1	yes	characteristic
332	C₂	#22569127	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1451	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1451	D₆₇
hypocenter	C₆₇	#18043
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22569227	C₂	semidirect
3. #22559124	C₂	semidirect
4. #22559089	C₂	semidirect
5. #22559029	C₂	semidirect
6. #22558972	C₂	semidirect
7. #22558644	C₂	semidirect
8. C₆₇×#1451	C₂	semidirect
83. Dic₂₆₈	M₁₆	semidirect
174. #1451	D₆₇	semidirect
175. C₆₇	#18043	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22569227	2
C₄×#14524065	2
#22559124	2
#22559089	2
#22559029	2
#22558972	2
#22558644	2
C₆₇×#1451	2
#18043	67